ARITHMETIC THE SECOND product of (flls az) and C61S &2) is (a^b^ a^b^ azbit ajbz). It is always possible to reduce any such product-symbol (for a field of degree n) to a symbol containing at most n integers. One final short remark completes the synopsis of the story. An ideal whose symbol contains but one integer, such as (GX), is called a principal ideal. Using as before the notation (a-^ ' (b^) to signify that (a-^ contains (b^, we can see without difficulty that (ax) ! (&i) when, and only when, the integer 0T divides the integer br As before, then, the concept of arithmetical divisi- bility is here - for algebraic integers - completely equivalent to that of class inclusion. A prime ideal is one which is not 'divi- sible by* - included in - any ideal except the all-inclusive ideal which consists of all the algebraic integers in the given field. Algebraic integers being now replaced by their corresponding principal ideals, it is proved that a given ideal is a product of prime ideals hi one way only, precisely as in the 'fundamental theorem of arithmetic' a rational integer is the product of primes in one way only. By the above equivalence of arith- metical divisibility for algebraic integers and class inclusion, the fundamental theorem of arithmetic has been restored to integers in algebraic number fields. Anyone who will ponder a little on the foregoing bare outline of Dedekmd's creation will see that what he did demanded penetrating insight and a mind gifted far above the ordinary good mathematical mind in the power of abstraction. Dedekind was a mathematician after Gauss' own heart: 'At nostro quidem juditio hujusmodi veritates ex notiombus polios quam ex nota- tionibus hauriri debeanf (But in our opinion such truths •[arithmetical] should be derived from notions rather than from notations). Dedekind always relied on his head rather than on an ingenious symbolism and expert manipulations of formulae to get hi™ forward. If ever a man put notions into mathematics, Dedekind did, and the wisdom of his preference for creative ideas over sterile symbols is now apparent although it may not have been during his lifetime. The longer mathematics lives tlie more abstract - and therefore, possibly, also the more practical - it becomes.